Finite projective planes are fascinating objects from many perspectives. In addition to the geometric view, they can be viewed as combinatorial block designs.

From the geometric perspective, there are two very important structural properties for projective planes: the Theorem of Desargues, which holds exactly when the plane can be coordinatized by a division ring, and the Theorem of Pappus, which holds precisely when the plane can be coordinatized by a field. It is a famous theorem of Wedderburn that every finite division ring is a field, so the two properties are equivalent for finite projective planes.

Although they are both very combinatorial statements, I don't recall having seen anything similar to Desargues and Pappus for other classes of block designs.

Are there interesting analogues or generalizations of the properties of Desargues and Pappus for other classes of block designs? Of particular interest would be analogues and generalizations that correspond (not necessarily exactly) to some form of coordinatization of designs.

Comment. This question has generated a fair amount of interest. I've been considering accepting the answer by John Conway and Charles Roque even though it is not quite satisfactory. (It only answers the first part of the question in a loose sense, and it does not address the second part.) So I decided to set up a small bounty to stimulate other answers.

1 Answer
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I passed on your question to John H. Conway. Here is his response: (NB. Everything following this line is from Conway and is written from his point of view. Of course, in the comments and elsewhere on the site, I am not Conway.)

I think it's wrong to focus on block designs in particular. This may not answer your question, but there are some interesting examples of theorems similar to Desargues's and Pappus's theorems. They aren't block designs, but they do have very nice symmetries.

I call these "presque partout propositions" (p.p.p. for short) from the French "almost all". This used to be used commonly instead of "almost everywhere" (so one would write "p.p." instead of "a.e."). The common theme of the propositions is that there is some underlying graph, where vertices represent some objects (say, lines or points) and the edges represent some relation (say, incidence). Then the theorems say that if you have all but one edge of a certain graph, then you have the last edge, too. Here are five such examples:

Desargues' theoremGraph: the Desargues graph = the bipartite double cover of the Petersen graphVertices: represent either points or linesEdges: incidenceStatement: If you have ten points and ten lines that are incident in all of the ways that the Desargues graph indicates except one incidence, then you have the last incidence as well. This can be seen to be equivalent to the usual statement of Desargues's theorem.

Pappus's theoremGraph: the Pappus graph, a highly symmetric, bipartite, cubic graph on 18 verticesVertices: points or linesEdges: incidenceStatement: Same as in Desargues's theorem.

"Right-angled hexagons theorem"Graph: the Petersen graph itselfVertices: lines in 3-spaceEdges: the two lines intersect at right anglesStatement: Same as before, namely having all but one edge implies the existence of the latter. An equivalent version is the following: suppose you have a "right-angled hexagon" in 3-space, that is, six lines that cyclically meet at right angles. Suppose that they are otherwise in fairly generic position, e.g., opposite edges of the hexagon are skew lines. Then take these three pairs of opposite edges and draw their common perpendiculars (this is unique for skew lines). These three lines have a common perpendicular themselves.

Roger Penrose's "conic cube" theoremGraph: the cube graph Q3Vertices: conics in the planeEdges: two conics that are doubly tangentStatement: Same as before. Note that this theorem is not published anywhere.

Standard algebraic examplesGraph: this unfortunately isn't quite best seen as a graphStatement: Conics that go through 8 common points go through a 9th common point. Quadric surfaces through 7 points go through an 8th (or whatever the right number is).

Anyway, I don't know of any more examples.

Also, I don't know what more theorems one could really have about coordinatization. I mean, after you have a field, what more could you want other than, say, its characteristic? (Incidentally, the best reference I know for the coordinatization theorems is H. F. Baker's book "Principles of Geometry".)

How does the proof go for the projective planes? I assume it's that you use the patterns of lines to perform arithmetic on the coordinates. So, can you perform arithmetic on coordinates via some of these other structures? If so, then hypergraphs with the same properties might be good candidates for structures which imply nice coordinate systems are possible, even if they don't turn out to be block designs.
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Douglas ZareJan 23 '10 at 7:20

I really appreciate this answer, I wasn't aware of Penrose's Conic Cube Theorem. Unfortunately, as you point out at the beginning, this doesn't really answer my question. The "standard algebraic examples" are what I had in mind as a basis for potential generalization to other classes of block designs (or, as Doug Zare suggests, classes of hypergraphs). I voted up your answer since the information is both relevant and interesting, but I cannot currently accept it since it doesn't really answer the question as posed.
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François G. Dorais♦Jan 23 '10 at 15:10

The first sentence "I think it's wrong to focus on block designs in particular" suggests Conway believes that 'p.p. properties' like Desargues and Pappus are in some way fundamentally geometric and therefore not combinatorial. I'm not sure if that is what he meant or, for that matter, exactly what that means. (By the way, it would be useful to clearly delineate Conway's contributions to the answer from yours to avoid confusion and possible misattribution.)
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François G. Dorais♦Jan 23 '10 at 15:20

Re Conway vs. me: everything following the first sentence is written from Conway's point of view. Notably, all "I"s are Conway. Re "I think it's wrong to focus on block designs in particular": he means that blocks designs are not the only possible interesting symmetries. That is, he thinks the cube graph is just as interesting as the block design in Desargues's theorem.
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aorqJan 23 '10 at 19:10

What do you mean when you compare the "block design in Desargues's Theorem" with the cube graph? A suitable set of conics could work just as well as lines to form a blocks of design. I don't know, but maybe Penrose's Theorem could be reinterpreted as such a "coordinatization theorem." (This is actually where I get hung up trying to figure out Conway's point of view.)
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François G. Dorais♦Jan 23 '10 at 22:35